Question

If $S=\sin \frac{\pi }{n}+\sin \frac{3\pi }{n}+\sin \frac{5\pi }{n}+\cdot \cdot \cdot n$ terms.
Then the value $nS$ is

Answer 1

Given series $S=\sin \frac{\pi }{n}+\sin \frac{3\pi }{n}+\sin \frac{5\pi }{n}+\cdot \cdot \cdot +\sin (\alpha +(n-1\left)\beta \right)$.

Comparing the given series with standard sine series

$\sin \alpha +\sin (\alpha +\beta )+\sin (\alpha +2\beta )+\sin (\alpha +3\beta )+\cdot \cdot \cdot$
Here,
$\alpha =\frac{\pi }{n}$
and $\beta =\frac{2\pi }{n}$

Sum of the sine series $=\frac{\sin \frac{n\beta }{2}}{\sin \frac{\beta }{2}}\sin (\alpha +\frac{(n-1)\beta }{2})$

$=\frac{\sin \left(\frac{n\times 2\pi }{2\times n}\right)}{\sin \left(\frac{2\pi }{2\times n}\right)}\sin (\frac{\pi }{n}+\frac{(n-1)2\pi }{2n})$

$=\frac{\sin \pi }{\sin \left(\frac{\pi }{n}\right)}\times \sin (\frac{\pi }{n}+\frac{(n-1)\pi }{n})$

$=\frac{0}{\sin \left(\frac{\pi }{n}\right)}\cdot \sin (\frac{\pi }{n}+\frac{(n-1)\pi }{n})$

$=0$

$n\cdot S=n\times 0=0$